Finite Group Databasebeta

← #27297314

Subgroups — #27297314

Order 13568.

344 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285490C211yesmaximal, characteristic
2#20276917C212yesmaximal, complemented
3#20272726C212yesmaximal, complemented
4#20272639C211yesmaximal, characteristic
5C53×#2076C211yesmaximal, characteristic
6C2×#13133196—21no—
7#16810959—22no—
8#16810864—21no—
9#16810864C2212yes—
10#16810781—22no—
11#16810448—22no—
12#16810426—22no—
13C22⋊#11719908C2212yes—
14C2×#107×C53C2211yescharacteristic
15C2×C22⋊C8×C53C2211yescharacteristic
16C53×#380—21no—
17C53×#340—21no—
18C53×#340C2211yescharacteristic
19C22×Dic106—21no—
20C2×Dic212—21no—
21C2×Dic212D412yes—
22#13133196—22no—
23#13133196—22no—
24#13133196—41no—
25#13133196—41no—
26C2×C4⋊Dic53—21no—
27C2×#11719919—41no—
28C2×C22⋊Dic53—41no—
29C2×#11719908—22no—
30#13133151—21no—
31#13133151D412yes—
32#13133142—22no—
33#13133142—22no—
34C23×C4×C53D411yescharacteristic
35C22×C8×C53D411yescharacteristic
36C22×C8×C53C2311yescharacteristic
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C22⋊C4×C53—41no—
40#107×C53—22no—
41C22⋊C8×C53—22no—
42#95×C53—41no—
43C24×C53D811yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53D811yescharacteristic
51C22×C4×C53C2×D411yescharacteristic
52C22×Dic53—41no—
53C22×Dic53—41no—
54C2×Dic106—22no—
55C2×Dic106—41no—
56C2×Dic106—81no—
57C2×Dic106—81no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53—41no—
61C2×C8×C53C2×D412yes—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—82no—
67C22⋊Dic53—42no—
68C22⋊Dic53—42no—
69C4⋊Dic53—22no—
70C4⋊Dic53—41no—
71#11719919—42no—
72#11719919—42no—
73Dic212—42no—
74Dic212—81no—
75#11719908—42no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—41no—
79C23×C53—41no—
80C23×C53Q3212yes—
81C23×C53C2×D811yescharacteristic
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—81no—
89C2×C4×C53—81no—
90C2×C4×C53—81no—
91C2×C4×C53—81no—
92C2×C4×C53—81no—
93C2×C4×C53—81no—
94C2×C4×C53—81no—
95C2×C4×C53#12011yescharacteristic
96C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—82no—
100C2×Dic53—82no—
101Dic106—42no—
102Dic106—42no—
103Dic106—82no—
104C8×C53—22no—
105C8×C53—41no—
106#12282—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—22no—
110C22×C53—42no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—81no—
114C22×C53#44711yescharacteristic
115C22×C53C2×Q3211yescharacteristic
116C22×C53#51011yescharacteristic
117C4×C53—21no—
118C4×C53—41no—
119C4×C53—161no—
120C4×C53—161no—
121C4×C53#45511yescharacteristic
122C4×C53#45811yescharacteristic
123Dic53—82no—
124Dic53—82no—
125#3189—531no—
126#2236—532no—
127#2076D5311yescharacteristic
128#1348—532no—
129#1314—531no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53—42no—
133C2×C53#160711yescharacteristic
134C2×C53#218811yescharacteristic
135C2×C53#219011yescharacteristic
136C2×C22⋊Q8—1061no—
137#484—1062no—
138#467—532no—
139#467—1061no—
140#451—1062no—
141C2×#107C2×D5311yescharacteristic
142C2×C22⋊C8C2×D5311yescharacteristic
143#380—21no—
144#366—1062no—
145#358—1062no—
146C22⋊C16—532no—
147#340—21no—
148#340C2×D5311yescharacteristic
149C53#1228211yescharacteristic, complemented, Sylow
150C22×Q8—1061no—
151C23×C4C22⋊D5311yescharacteristic
152C2×Q16—532no—
153C2×Q16—1061no—
154C22×C8C22⋊D5311yescharacteristic
155C22×C8C22×D5311yescharacteristic
156C22⋊Q8—1062no—
157C22⋊Q8—1062no—
158C22⋊Q8—2121no—
159C22⋊Q8—2121no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—2121no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—2121no—
166C2×C16—1062no—
167#107—22no—
168#107—532no—
169#107—1061no—
170#103—1062no—
171#103—1062no—
172C22⋊C8—22no—
173#95—41no—
174C24#1171992211yescharacteristic
175C2×Q8—1062no—
176C2×Q8—4241no—
177C2×Q8—4241no—
178C2×Q8—2121no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—2121no—
186C22×C4—2121no—
187C22×C4#1171992211yescharacteristic
188C22×C4C2×C22⋊D5311yescharacteristic
189Q16—4241no—
190Q16—2122no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8D4×D5312yes—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—1062no—
200C4⋊C4—2121no—
201C4⋊C4—2122no—
202C4⋊C4—2122no—
203C22⋊C4—82no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C16—2122no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1313316312yes—
212C23C2×#1171992211yescharacteristic
213Q8—4242no—
214Q8—2122no—
215Q8—2122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—4242no—
230C2×C4—4242no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4#1313318511yescharacteristic
234C2×C4#1313324211yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1681075611yescharacteristic
245C22C2×#1313316311yescharacteristic
246C22#1681106311yescharacteristic
247C4—21no—
248C4—41no—
249C4—161no—
250C4—161no—
251C4—4242no—
252C4—4242no—
253C4#1681080011yescharacteristic
254C4#1681081211yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2027370011yescharacteristic
259C2#2027664811yescharacteristic
260C2#2027666011yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2076C2
socleC22×C53#447
prime radicalC24×C53D8
hypercenter#2076D53
hypocenterC53#12282
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2076

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20276917C2semidirect
3. #20272726C2semidirect
149. C53#12282semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202854902
#202769172
#202727262
#202726392
C53×#20762
#1228253